Explicit q-action making the thimble isomorphism an [q]-module isomorphism

Determine the explicit [q]-module structure on H^*(M)[q] ⊕ ⊕_{w≥1} H^*(D) z^w that makes the thimble map H^*(M)[q] ⊕ ⊕_{w ≥ 1} H^*(D) z^w → SH^*_q(M,D) from Theorem 1.1 an isomorphism of [q]-modules; characterize this action in terms of enumerative invariants, presumably punctured Gromov–Witten invariants of the pair (M,D).

Background

The main theorem provides a canonical isomorphism SH^*_q(M,D) ≅ H^*(M)[q] ⊕ ⊕_{w≥1} H^*(D) z^w and notes that the [q]-module structure extends uniquely to make this map an isomorphism. However, the authors do not describe this [q]-action explicitly and suggest that its detailed form is expected to be governed by punctured Gromov–Witten theory.

Clarifying the [q]-action would illuminate how the deformation by allowing intersections with the anticanonical divisor D integrates enumerative data and would strengthen the link between deformed symplectic cohomology and relative Gromov–Witten invariants.